时空弹性带方法：机器人高效轨迹优化

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"这篇论文《2013_Efficient Trajectory Optimization using a Sparse Model》是2013年关于机器人路径优化的研究，采用稀疏模型提高效率，属于计算机科学与机器人学领域的专业论文。" 正文：这篇论文的核心是介绍了一种名为“时间弹性带（Timed Elastic Band, TEB）”的高效轨迹优化方法，该方法主要用于机器人路径规划。时间弹性带技术通过逐步修改由全局规划器生成的初始路径来优化机器人的运动轨迹，同时考虑了多种目标因素，包括但不限于总路径长度、执行时间、与障碍物的安全距离、经过中间目标点以及遵循机器人的动态、运动学和几何约束。在时间弹性带的方法中，路径规划不仅考虑空间上的路径，还强调了时间上的动态约束，如机器人的最大速度和加速度限制。这一特性使得算法能在实时环境中运行，能够适应动态变化的障碍物和运动限制。论文将时间弹性带问题转化为一个标量化多目标优化问题，这允许算法处理多个相互冲突的目标，如时间和距离的平衡，同时保持路径的安全性。论文指出，大多数优化目标是局部的，只与连续的一小部分机器人状态有关。这种局部结构是算法效率的关键，因为它允许算法快速地针对特定参数进行调整，而不需要对整个路径进行大规模的计算。通过利用这种稀疏模型，可以显著减少计算复杂度，提高实时优化的性能。此外，论文可能还深入探讨了如何构建和求解这个优化问题的具体算法，可能包括线性化技术、近似策略和迭代过程。论文可能会讨论如何处理不完全或不确定的信息，例如对障碍物的动态预测，以及如何在优化过程中集成这些不确定性。这篇2013年的研究为机器人路径规划提供了一种创新且高效的解决方案，它强调了实时性和动态适应性，对于自动驾驶、服务机器人和其他需要实时路径优化的领域具有重要的理论和应用价值。通过深入理解时间弹性带方法，开发者可以设计出更智能、更快速响应环境变化的机器人系统。

Efﬁcient Trajectory Optimization using a Sparse Model

Christoph R

osmann

, Wendelin Feiten

, Thomas W

osch

, Frank Hoffmann

and Torsten Bertram

Abstract— The ”timed elastic band” approach optimizes

robot trajectories by subsequent modiﬁcation of an initial tra-

jectory generated by a global planner. The objectives considered

in the trajectory optimization include but are not limited to

the overall path length, trajectory execution time, separation

from obstacles, passing through intermediate way points and

compliance with the robots dynamic, kinematic and geometric

constraints. ”Timed elastic bands” explicitly consider spatial-

temporal aspects of the motion in terms of dynamic constraints

such as limited robot velocities and accelerations. The trajectory

planning operates in real time such that ”timed elastic bands”

cope with dynamic obstacles and motion constraints. The

”timed elastic band problem” is formulated as a scalarized

multi-objective optimization problem. Most objectives are local

and relate to only a small subset of parameters as they

only depend on a few consecutive robot states. This local

structure results in a sparse system matrix, which allows the

utilization of fast and efﬁcient optimization techniques such

as the open-source framework ”g2o” for solving ”timed elastic

band” problems. The ”g2o” sparse system solvers have been

successfully applied to VSLAM problems. This contribution

describes the application and adaptation of the g2o-framework

in the context of trajectory modiﬁcation with the ”timed

elastic band”. Results from simulations and experiments with a

real robot demonstrate that the implementation is robust and

computationally efﬁcient.

I. INTRODUCTION

Trajectory planning ﬁnds an optimal collision free trajec-

tory that complies with the robots kinematic and dynamic

motion constraints. This paper focuses on trajectory modi-

ﬁcation assuming that a global planner generated an initial

feasible path beforehand [1]. In particular in the context of

service robotics the dynamic modiﬁcation of a preplanned

path is preferable over ofﬂine trajectory planning. Online

modiﬁcation copes with changes of a dynamic environment

by incorporating the most recent sensor data for local re-

ﬁnement of the trajectory. In most realistic applications the

model of the environment is subject to continuous change

due to partial, incomplete maps and dynamic obstacles.

Furthermore, the (re-)computation of a large scale global

path is often not feasible in real-time applications. This

observation leads to approaches which modify a path locally,

such as the ”elastic band” proposed by [2], [3].

Later the original approach was extended to nonholonomic

kinematics [4], [5], [6] and robotic systems with many

degrees of freedom [7]. [8] proposed a method, where an

initial path is deformed using optimization techniques. The

trajectory, i.e. the velocities along the path, are not optimized.

The time parameter is used to control the modiﬁcations of

Institute of Control Theory and Systems Engineering, Technische Uni-

versit

at Dortmund, Germany

Siemens Corporate Technology, Research Group Robotics, Germany

the path as the optimization proceeds. The planner considers

the nonholonomic constraints.

[9] deals with the deformation of trajectories rather than

paths by an explicit consideration of temporal information.

The deformation is decomposed into an obstacle avoidance

step using repulsive forces and a connectivity maintenance

step. Based on this work [10] proposes a single step approach

that combines external deformation with internal connec-

tivity forces. Both methods support general state transition

models and allow for spatial-temporal obstacle avoidance.

In contrast, our approach is based on a graph-optimization

formulation, operates with general optimization solvers and

time optimality is an explicit objective. Other methods which

directly optimize trajectories are presented in [11], [12].

In our case, a parametric path is augmented with velocity

proﬁles that respect the kinodynamic constraints of the

platform. The approach starts with an initial path found by

a global planner and represents it by a compact spline-based

path model also used in [13]. This path model exposes a set

of higher-level parameters to the optimization that iteratively

adapts the shape of the curvature continuous path to reduce

an objective function such as the time of travel. The main

difference to our work is that it trades in the precision of the

analytic model for a discretized trajectory model that allows

it to employ a highly sophisticated, efﬁcient optimization

algorithm, enabling trajectory reﬁnement in real-time.

Most recent approaches for trajectory modiﬁcation dealing

with robot arms with many degrees of freedom use a

discretized representation of the trajectory in conﬁguration

space (see [15], [16]). The proposed objective function

contains a ﬁnite difference matrix to smooth the resulting

trajectory and to additionally satisfy constraints like obstacle

avoidance. The CHOMP algorithm relies on a covariant

gradient descent method which explicitly requires the gra-

dients of each objective, whereas STOMP uses a stochastic

trajectory optimization technique without explicit knowledge

of gradients. Both approaches include temporal information

only in implicit manner by deﬁning a speciﬁc discretization

and task duration. The differences to our approach are

detailed in Section IV.

In [17] the authors introduced a new approach called

”timed elastic band” which explicitly augments ”elastic

band” with temporal information. The proposed extension

allows the consideration of the robot’s dynamic constraints

and direct modiﬁcation of trajectories rather than paths.

The ”timed elastic band” is formulated as a scalarized

multi-objective optimization. The structure of the underlying

optimization problem is sparse as most objectives are local

in that they only depend on a few consecutive conﬁgurations

2013 European Conference on Mobile Robots (ECMR)

Barcelona, Spain, September 25-27, 2013

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